Weighted Sobolev spaces and capacity.

Abstract

Let Ω be an open set in R and 1 < p < ∞ . In this paper we consider the theory of weighted Sobolev spaces H with weight function in Muckenhoupt’s Ap -class. Our main purpose is to provide a coherent exposition of the behavior of functions in weighted Sobolev spaces and this leads us to use a concept of capacity. The motivation arises from the theory of partial differential equations, see e.g. [F], [HKM]. Most of the results we present are probably not new but according to our knowledge they have not yet appeared in printed form. We define the weighted Sobolev space H(Ω;w) to be the completion of C(R) with respect to the norm